Step 1
^{Group Information}

Calculate the geometric mean of information from the proportionality of \(B^\dagger_i\) (\(\omega_z\) below) conditioned by group (subscript \(i\)) and within-group break cycle \(B_i\) ^{Eq. 3ba}:

\[ I_i := \Big[\Pi_{q = 1}^z - \log_2 (\omega_q \enspace |\enspace B = B_i)\Big]^{\frac{1}{z}} , \enspace 0 < \omega_z \le 1 \longrightarrow \big<I_1 \enspace I_2\enspace\dots\enspace I_z\big>_i \\ \enspace \\ \omega_i := \big|\big|\chi_z\big|\big|^{-1} \bigg( {\sum_{q = 1}^z \chi_q} \bigg) \longrightarrow \big<\omega_1 \enspace \omega_2 \enspace \dots \enspace \omega_z \big>_i \\ \enspace \\ \chi_{i} := \beta^\dagger_z \enspace | \enspace (\Phi_q \le \delta_q,\enspace B_i) \Big|_{q = 1}^z \longrightarrow \big<\chi_1\enspace\chi_2\enspace\dots\enspace\chi_z\big>_i \]

Step 2
^{Information PMF}

Separately calculate the geometric (negative binomial with \(r = 1\)) probability mass function (\(g_i\)) over \(P(K_N)\) ^{Eq. 4} by within-group cycle (\(B_i\)):

\[ g_i := p_i (1 - p_i)^{k_i} \longrightarrow \big<g_1\enspace g_2\enspace \dots\enspace g_z\big>_i\\ \enspace \\ p_i:= n^{-1} \sum_{q = 1}^{n = ||\Phi_i||} \Big\{P(K_N) \enspace |\enspace K_N\cap \Phi_i \Big\}_{q\le n} \longrightarrow \big<p_1\enspace p_2\enspace \dots\enspace p_n\big>_i \\ \enspace \\ k_i:= \frac {\sum_{q = 1}^j \phi_{j \le q} } {\sum_{q = 1}^j \phi_q}, \enspace \phi_j \in \Phi_i \longrightarrow \big<x_1\enspace x_2\enspace \dots\enspace x_z\big>_i, \enspace k_i \in B^\dagger_i \]

In the preceding equation, for each within-group cycle ^{Eq. 3a}:

• \(p_i\) is a cumulative proportionality of \(P(K_N)\) conditioned by each \(\Phi_i\)

• \(k_i\) is a cumulative summation of \(\Phi_i\). Since the \(g_i\) returns the probability of \(k\) attempts before the first success, the cumulative summation of each \(\Phi_i\) serves as a proxy for an “attempt” (which is why the original data must be ordered before calculating \(\Phi\)). In this regard, \(k_i\) operates under a similar context as survival time .

The consequence is that \(g_i\) is a vector of draws from the geometric PMF conditioned by each \(\Phi_i\) and (implicitly) \(B_i\). Over all groups and their respective cycles, what one ends up with is a likelihood space mapped to cycle duration and the sequential breaks within each cycle.

Step 3
^{Information Measures & Score}

Calculate the squared deviations of within-group cycle information \(I_i\) ^{Step 1} from global information \(I_N\) ^{Eq. 5} conditioned on \(\Phi_i\) and the associated scoring vector \(K^\dagger_i\) (defined below):

\[ \Theta_i := \big[(I_N \enspace \big|\enspace K_N \cap \Phi_i \ne \emptyset) - I_i \big]^2 \longrightarrow \big< \theta_1 \enspace \theta_2 \enspace \dots \enspace \theta_n \big>_i \]

\[ K^\dagger_i:= g_i \Theta_i \longrightarrow \big< \kappa_1 \enspace \kappa_2 \enspace \dots \enspace \kappa_n \big>_i \]

Next, calculate the effective slope and curvature of \(\Theta_i\) for each within-group cycle:

\[ \Theta_i^{'} := \frac{d}{dn}\theta_n\enspace\big|\enspace B_i \]

\[ \Theta_i^{''}:= \frac{d^2}{dn}\theta_n\enspace\big|\enspace B_i \]

Step 4
^{Scoring, Mean Info, and Derivatives}

\(\mathbb{D}\) is filtered by row where the following is true:

\[ k_i \le k_{max} \enspace\wedge\enspace \Theta^{'} \ge 0\enspace\wedge\enspace \Theta^{''} \le 0 \]

The result is augmented by deriving an information-based score (\(\Omega\)) as well as the weighted-mean information (\(\bar{I}\)) of \(\Theta_i\) conditioned on \(K_N\).

Given q: \(q \in \big\{1, 2, \dots, N\big\}\):

\[ \enspace \\ \Omega^k_R := \Big(K^\dagger \enspace | \enspace \big\{g_i = \text{max}_{g_i},\enspace K_q = k_R \big\} \Big) \longrightarrow \big<\omega_1\enspace\omega_2\enspace\dots\enspace\omega_R\big>^k \] {#eq:break-information-score} \[ \bar I_R^k:= \frac{\sum_{q = 1}^N{g_q}{I_R}}{\sum_{q = 1}^N g_q} \enspace\|\enspace K_q = k_R \longrightarrow \big<i_1\enspace i_2\enspace\dots\enspace i_R\big>^k \]

Note: the superscript \(k\) is an indicator that the metric is indexed by values in \(k\) — it is not an exponent.

Next, the slope and curvature of \(\Omega_R^k\) and \(\bar I_R^k\) with respect \(k\) are derived which maps the scores and mean information to values in \(k\):

\[ \frac{\partial}{\partial{k}} \big\{\Omega_R^k;\enspace \bar I_R^k\big\} \Rightarrow \big\{{\Omega_R^k}^{'};\enspace {{\bar I_R^k}}^{'}\big\} \enspace \\ \frac{\partial}{\partial{k}} \big\{\Omega_R^k;\enspace \bar I_R^k\big\} \Rightarrow \big\{{\Omega_R^k}^{''};\enspace {{\bar I_R^k}}^{''}\big\} \]

Step 5
^{Within-fold Scores, Optimal and Alternate Breaks}

The final scoring vector is defined as follows:

\[ \Gamma_R^k:= -\lambda({\bar I_R^k}^{''}) \times \lambda({\Omega_R^k}^{''}) \enspace\Big|_{q = 1}^R \longrightarrow \big<\gamma^k_1\enspace\gamma^k_2\enspace\dots\enspace\gamma^k_q\big>\enspace\big|_{q=1}^R\\ \lambda:= f(x)\to \frac {x}{x_{max}} \]

The optimal break (\(k\)) is the one that maximizes \(\Gamma_R^k\) by maximizing \({\Omega_R^k}^{''}\) and minimizing \({\bar I_R^k}^{''}\), with alternatives as those values where the score is some distance \(\zeta\) from the maximized \(\Gamma_R^k\):

\[ \hat{k} :=k\enspace|\enspace\gamma_q^k = \Gamma_{max} \] \[ \ddot{k}:=k\enspace|\enspace\gamma_q^k \ge \big(\Gamma_{max}-\zeta\big) \]

Step 6
^{Final Score, Optimal and Alternate Breaks}

Over \(\text{n_folds}\) of generated and scored data, we have \(k_n:=\big<k_1\enspace k_2\enspace \dots k_n \big>\) from which the optimal \(k\) is “greedily” chosen as the most frequently-occurring \(k_n\): \(\text{arg}_{_k}\text{max}\enspace{\#}k_n\big|_{q = 1}^n\).